Probability Matrix Research Articles

Measuring ecological effects and policy mechanisms of urban land use transition – a case study of Xi'an, China

ABSTRACT Measuring ecological effects and policy mechanisms of urban Land Use Transition (LUT) is the key to understanding regional environmental changes and achieving sustainable urban developments. This paper illustrates and demonstrates a formal method for measuring the ecological effect of LUT characterized by Ecological Security Index (ESI) and Ecosystem Value (ESV), and enhanced with a correlational analysis between the ecological effect model and urban policy mechanism. Using Xi'an, a major capital city of western China as a case study, based on multi-temporal Thematic Mapper (TM) remote sensing images, Digital Elevation Model (DEM) and relevant statistical data, the results show that the method using ESI and ESV, together with the Land Use Transition Probability Matrix (LUTPM) can comprehensively measure the ecological effects of LUT process in Xi'an from 2000 to 2020. The results show that due to the impact of urban expansion, regional construction land increased significantly (by 67.07%), while farmland decreased considerably (by 13.12%). The farmland and forestland around the main urban area of Xi'an were centrally converted into construction land, yet the mutual transition of other types of land was relatively scattered. The total amount of ESV has slightly increased from 40.1 billion yuan to 42.8 billion yuan, and the proportion of ecological service function value presents a decreasing trend in regulation function (65%), support function (22%), supply function (9%), and cultural function (4%). The ESI value slightly decreased from 0.517 to 0.512, and overall it is still in a state of ecological security. The trend of ESV and ESI decreasing from mountainous areas to plains indicates that high-intensity human activities in plain areas have had a significant impact on the ecological environment. The correlation analysis between policy mechanisms and ecological effects indicates that ecological protection policies have improved the ecological environment and maintain ecological security (high ESV-high ESI mode), and may also have limited economic activities and been detrimental to the security of urban composite ecosystems (high ESV-low ESI mode). Spatial expansion policies have driven the rapid growth of construction land and threatened ecological security (low ESV-low ESI mode). Zoning adjustment policies have led to the transformation of ecological land to construction land, threatening ecological security and reducing ESV (low ESV-low ESI mode), and may also have promoted industrial upgrading and economic development, providing economic assurance for improving ecological security (low ESV-high ESI mode). These comprehensive findings provide valuable insights for understanding the ecological effects and policy mechanisms of LUT in Xi'an for achieving sustainable urban development, and demonstrate and explore a novel method to more appropriately and effectively measure LUT by combining ESV and ESI.

Application of a surrogate model for condition monitoring of a digital twin gas turbine

Condition monitoring technology plays a crucial role in ensuring the reliable operation of gas turbines. Digital twin has propelled condition monitoring research into a new phase. This paper established a surrogate model of gas turbines for condition monitoring based on Markov-projection approximation subspace tracking. Furthermore, it explores the application of surrogate model in developing digital twin for gas turbines. The study initially establishes a Markov matrix and acquires an observation vector, utilizing the framework of the linear model. Utilizing real-time measurement data of gas turbine, the signal subspace of the observation vector autocorrelation matrix is updated through the projection approximation subspace tracking. By aligning this signal subspace with the generalized observability matrix, the identification results of the surrogate model parameters are obtained online. Furthermore, a variable weight projection approximation subspace tracking method has been proposed to enhance the algorithm robustness. Simulation and real experiment demonstrate that the surrogate model output effectively tracks the real-time changes in gas turbine measurement data. When faults and degradation arise, condition monitoring can be achieved by analyzing the evolution of model parameters to obtain feedback information from the gas turbine. The proposed method maintains its robustness in the presence of impulsive noise. These features offer a novel approach for the development of gas turbine digital twin.

Identifying loci in mobility networks with applications in New Zealand work commutes: a statistical test for identifying extreme stationary distribution values in Markov transition matrices

Human mobility describes physical patterns of movement of people within a spatial system. Many of these patterns, including daily commuting, are cyclic and quantifiable. These patterns capture physical phenomena tied to processes studied in urban planning, epidemiology, and other social, behavioral, and economic sciences. This paper advances human mobility research by proposing a statistical method for identifying locations that individual move to and through at a rate proportionally higher than other locations, using commuting data for the country of New Zealand as a case study. We term these locations mobility loci and they capture a global property of communities in which people commute. The method makes use of a directed-graph representation where vertices correspond to locations, and traffic between locations correspond to edge weights. Following a normalization, the graph can be regarded as a Markov chain whose stationary distribution can be calculated. The proposed permutation procedure is then applied to determine which stationary distribution values are larger than what would be expected, given the structure of the directed graph and traffic between locations. The results of this method are evaluated, including a comparison to what is already known about commuting patterns in the area as well as a comparison with similar features.

KPZ scaling from the Krylov space

Recently, a superdiffusion exhibiting the Kardar-Parisi-Zhang (KPZ) scaling in late-time correlators and autocorrelators of certain interacting many-body systems, such as Heisenberg spin chains, has been reported. Inspired by these results, we explore the KPZ scaling in correlation functions using their realization in the Krylov operator basis. We focus on the Heisenberg time scale, which approximately corresponds to the ramp–plateau transition for the Krylov complexity in systems with a large but finite number degrees of freedom. Two frameworks are under consideration: i) the system with growing Lanczos coefficients and an artificial cut-off, and ii) the system with the finite Hilbert space. In both cases via numerical analysis, we observe the transition from Gaussian to KPZ-like scaling, or equivalently, the diffusion-superdiffusion transition at the critical Euclidean time tE∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {t}_E^{\\ast } $$\\end{document} = ccrK, for the Krylov chain of finite length K, and ccr = O(1). In particular, we find a scaling ~ K1/3 for fluctuations in the one-point correlation function and a dynamical scaling ~ K−2/3 associated with the return probability (Loschmidt echo) corresponding to autocorrelators in physical space. In the first case, the transition is of the 3rd order and can be considered as an example of dynamical quantum phase transition (DQPT), while in the second, it is a crossover. For case ii), utilizing the relationship between the spectrum of tridiagonal matrices at the spectral edge and the spectrum of the stochastic Airy operator, we demonstrate analytically the origin of the KPZ scaling for the particular Krylov chain using the results of the probability theory. We argue that there is interesting outcome of our study for the double scaling limit of matrix models. For the case of topological gravity, the white noise term of order O (1/N) is identified, which should be taken into account in the controversial issue of ensemble averaging in 2D/1D holography.

Seismic vulnerability assessment of urban areas made of adobe buildings through analytical and numerical methods: The case study of the historical center of Cusco (Peru)

This paper aims at giving a contribution to the topic concerning the preservation of valuable adobe buildings, by setting a methodology for assessing their seismic vulnerability at the urban scale by means of analytical and numerical models, accounting for both their in-plane and out-of-plane behaviors. To this purpose, the valuable historical center of Cusco (Peru) is considered as representative case study. The city center of Cusco is an extraordinary example of colonial architecture that preserves essential aspects of its origins, it being representative of a large plethora of historical centers in South America and the Caribbean, where most of the residential buildings are made of adobe masonry.First, archetypes, representative of the whole historical center, are identified in order to get a manageable stock of buildings for which the analyses can be addressed to. The structural capacity of each archetype is then given in terms of capacity curves for both out-of-plane and in-plane mechanisms. Subsequently, these curves are compared with the likely seismic demands to assess the probability of attaining pre-established damage states for different seismic intensities. The obtained probabilities are collected into Damage Probability Matrices (DPMs) and, finally, cumulated to plot the fitting fragility curves.

Weighted-Neighborhood-Information-Network-Enabled Anomaly Detection Method for Electronic Sensors and Sensor Networks

As electronic sensors and sensor networks advance, perception data are increasingly characterized by mixed attributes. Traditional anomaly detection methods predominantly focus on numerical attributes. In this paper, we introduce a weighted neighborhood information network (WNIN)-enabled anomaly detection method tailored for mixed-attribute data from electronic sensors and sensor networks. Firstly, we employ the analytic hierarchy process (AHP) to analyze the security of sensor networks, leveraging a hierarchical electronic sensor network model to construct a hierarchical perception security architecture for anomaly detection. Subsequently, a neighborhood information system is established to ascertain the relationships between data objects with mixed attributes. We then develop the WNIN to encapsulate the relationships, and a state-transferring probability matrix based on data object similarity is derived. Ultimately, a random wandering process within the WNIN is executed, and the importance of data objects is evaluated using the steady-state distribution vector, thereby determining the anomaly data. Simulation outcomes reveal that our proposed method attains superior anomaly detection rates compared with existing methods.

Performance and trade competitiveness of Indian spices: A Markov Chain Approach

A study was conducted to examine the growth performance and pattern of spice crop export destinations in India. The growth rate, balance of trade and transitional probability matrix were applied to analyse the study by utilizing the past ten years’ data (2013-2022) collected from various national and international sources. The top five exported spices were selected based on their export values in 2021. The findings showed that India secured a huge surplus of ?2118426 lakhs in 2022 with the growth performance of exports (11.66 per cent) and imports (17.65 per cent) in the last ten years. Sri Lanka was the most stable country, with the highest retention capacity (86.12 per cent) for importing pepper. In contrast, China imported cumin and oleoresins (spice oil) with 86.32 and 47.04 per cent, respectively, and Germany imported mint (38.71 per cent) and Bangladesh for turmeric (74.87 per cent) from India. The study recommends that India should formulate a policy with the aim to create favourable trade relations with targeted countries, which would be a fruitful effort for earning foreign exchanges for the nation. Keywords: Export, import, Markov chain, spices, transitional probability matrix. JEL Codes: R11, P45, Q13, Q17.

Information-directed policy sampling for episodic Bayesian Markov decision processes

We consider finite-stage Markov Decision Processes (MDPs) under incomplete information, where the decision-maker only knows that the true transition probability and reward matrices belong to given, finite sets. The decision-maker interacts with the system over a finite number of episodes. The first episode begins with a probabilistic belief about the true probability and reward matrices. This belief is updated at the end of each episode using observed events. The goal is to maximize the expected total reward earned over all episodes. In the resulting model-based episodic Bayesian MDP, it suffices to only consider (the known) policies that are optimal to each one of the possible probability and reward matrices. Nevertheless, the decision-maker should execute policies that provide information about the true probabilities and rewards (exploration), but also exploit this knowledge to increase rewards. We propose a framework called Information-Directed Policy Sampling (IDPS). In each episode, the decision-maker balances the exploitation-exploration trade-off by executing a randomized policy that minimizes a so-called convex information ratio. We derive a regret bound that is independent of state- and action-space cardinalities when the set of matrices is exogenously determined. Numerical experiments show IDPS outperforming a state-of-the-art approach called Posterior Sampling.

Effect of interchange spacing on drivers' visual characteristics in interchange merging areas

To explore the effect of interchange spacing on drivers' visual characteristics in the merging areas of interchange, a high-density group of five interchanges on the expressway of Chongqing, China, was selected as the test site. An naturalistic driving test was conducted with 47 participants, and the Tobii Glasses II portable eye tracker was used to collect gaze data during driving. The drivers' fixation field was divided into six areas by applying a K-means dynamic clustering algorithm combined with the actual scenario. Markov chains were used to calculate the drivers' gaze transition probability matrices under different driving conditions, and the analysis of gaze transition behaviors was directed at common spacing interchanges, small spacing interchanges, and composite interchanges. Under the ramp-mainline condition, drivers’ fixations were primarily concentrated on the near ahead and the left side areas, with higher rates of repeated fixations on the left rearview mirror and left-side line areas. The average value of fixation duration, saccade distance, and saccade speed of small spacing interchange is higher than common spacing interchange. Additionally, under the mainline condition, the probability of one-step transition and repeated fixation rates significantly increased for the right-side lane areas, and the average values of fixation index and saccade index of small spacing interchange are lower than those of common spacing interchange. The results show that the highest probabilities of repeated fixation by drivers occurred in the near ahead and far ahead areas in the interchange merging areas. Insufficient spacing resulted in more frequent occurrences of zero values in one-step transition probability matrices. The research conclusions provide theoretical support for the optimal design and safe operation of the merging area of high-density interchange group of urban expressway.

Carbon metabolism mechanisms and evolution characteristics analysis of the food-water-energy nexus system under blue-green infrastructure changes

Food, water, and energy comprise a complex system (FWE nexus) that generates much carbon emissions during operation. At the same time, urban blue-green infrastructure (BGI) has a critical carbon sequestration function. This paper combines the functions of the FWE nexus and BGI and uses ecological network analysis (ENA) and the Markov model to measure the carbon metabolism (CM) mechanisms and evolutionary characteristics of BGI and FWE nexus (BGI-FWE nexus) complex systems. The results show that Guangzhou has high carbon emissions, and Zhaoqing and Huizhou have high carbon sequestration. Resident land and industrial and transportation land transfers to different land uses are more likely to produce positive carbon flows, while BGI transfers to other types are more likely to produce negative carbon flows. The study of CM mechanisms reveals a high proportion of competition relationships and a low proportion of mutualism relationships. The ecological utility index (EUI) tends to fall initially and then increase, peaking at 0.84 in 2015–2020, the highest value for the study period. The CM network has less system robustness (SR) and is in an unsustainable state of high redundancy and low efficiency. The mechanism evolution characterization study’s findings show a decreased likelihood of remaining original and less stability in the spatial transfer probability matrices of EUI and SR. In this study, we constructed a BGI-FWE nexus research framework based on the different CM functions of BGI and FWE nexus. The research framework contributes to the realization of carbon reduction in the FWE nexus system and is essential for the planning and management of urban BGI.

Regional Differences, Dynamic Evolution, and Convergence of Global Agricultural Energy Efficiency

Understanding the regional disparities, dynamic evolution, and convergence–divergence characteristics of global agricultural energy efficiency is crucial for enhancing agricultural energy efficiency, ensuring food security, and responding to global green development trends. This paper utilizes 2002–2021 panel data from 144 countries globally, employing the epsilon-based measure–global Malmquist–Luenberger (EBM-GML) model to estimate agricultural energy efficiency, considering unexpected output. The Dagum Gini coefficient, kernel density estimation, spatial Markov matrix, and spatial convergence model are employed to explain the spatial patterns and evolving trends of global and regional agricultural energy efficiency at three levels: regional disparities, dynamic evolution, and convergence. The results indicate significant spatial heterogeneity in global agricultural energy efficiency, with Europe exhibiting the highest efficiency, followed by Asia and the Americas, while Oceania and Africa demonstrate the lowest efficiency. Agricultural energy efficiency globally and in each region continues to improve, with increasing regional disparities, and difficulties in grade transitions in agricultural energy efficiency across regions. Each region exhibits β-convergence characteristics, but the convergence rates vary, and various factors influence growth rates of agricultural energy efficiency differently across regions. Therefore, countries should tailor their strategies based on local conditions, considering their own resource endowments and developmental stages, and strengthen international exchanges and cooperation.

Research on the relationship of coupling coordination between digitalization and green development

The coupling coordination of digitalization and green development has become an inevitable requirement for building a new development pattern and achieving high-quality development of China’s economy. Based on the panel data of 284 Chinese cities from 2011 to 2021, this study uses the coupling coordination degree model, Dagum Gini coefficient, spatial convergence, Markov transfer probability matrix, and panel Tobit model to quantitatively analyze the spatial–temporal evolution characteristics and influencing factors of the coupling coordination degree of digitization and green development. The study results show that the coupling coordination degree (CCD) of digitalization and green development has an overall increasing trend during the study period, and the eastern region is higher than other regions which showing spatial non-equilibrium characteristics. The spatial difference of the CCD continues to downward, and inter-regional differences are the main source of the CCD. In the long run, there is a “catch-up effect” between cities with low CCD and those with high CCD, and the CCD of digitalization and green development will tend to a steady state. The transfer type of the CCD has the characteristics of “club convergence”, the probability of maintaining the original state is high, and the overall CCD shows a good development trend in the future. Factors such as environmental regulation, green innovation, industrial structure upgrading and financial efficiency can significantly contribute to the CCD of digitalization and green development. The above findings provide empirical evidence of the CCD of digitalization and green development to achieve high-quality economic development.

Research on the strategy for average consensus control of flywheel energy storage array system based on lifecycle

In the domain of clean energy, the flywheel energy storage array system (FESAS) is widely employed for efficient and renewable energy storage to stabilize power grids and smooth power variations. However, managing the lifecycle of FESAS and achieving coordinated control presents a considerable challenge. This study integrates a lifecycle parameter based on the average consensus algorithm, enabling flywheels with longer lifecycles to handle additional charging and discharging tasks, while those with shorter lifecycles participate less. This approach effectively balances flywheel lifecycles and extends the overall lifespan of FESAS. Experimental findings reveal that when the P matrix satisfies the condition of equal row sums or column sums, the power distribution parameters tend toward average consensus, enabling FESAS to efficiently smooth unbalanced power. In undirected graphs, the symmetry of the P matrix causes the auxiliary variables to tend toward average consensus, effectively suppressing unbalanced power. For directed unbalanced graphs, this study proposes Error Balancing Iterative Algorithm (EBTA) to construct a doubly stochastic matrix P, facilitating the convergence of auxiliary variables to average consensus and effectively suppressing unbalanced power. The experimental results validate the efficacy of our proposed approach. This research establishes a theoretical foundation for the intelligent coordinated control of FESAS, highlighting the pivotal role of lifecycle parameters, and offers valuable insights for enhancing the performance and lifespan of FESAS.

Inclusion of fibrinoid necrosis increases the accuracy of synovial tissue assessment in predicting response to methotrexate: analysis of the UCLouvain Brussels ERA Cohort

ObjectiveRheumatoid Arthritis (RA) often exhibits suboptimal treatment response despite early diagnosis and treatment. This study aimed to analyze Early Rheumatoid Arthritis (ERA) synovial biopsies through histology and immunohistochemistry (IHC) to identify predictive factors for treatment response to Methotrexate (MTX).Methods140 ERA patients from the UCLouvain Arthritis Cohort underwent synovial biopsy and were monitored after initiating Disease-Modifying Antirheumatic Drug (DMARD) therapy. Histological features [Synovial Hyperplasia, Fibrinoid Necrosis (FN), Hypervascularization and Inflammatory Infiltrate] and IHC (CD3, CD20, CD138, CD68) were each semi-quantitatively assessed on a 0–3 scale with 7 levels.ResultsA strong association was observed between synovial CD68 and Fibrinoid Necrosis scores [r = 0.44 (0.27 − 0.56); p < 0.0001]. CD68 correlated with C-Reactive Protein (CRP), DAS28, SDAI and CDAI. Fibrinoid Necrosis score correlated with CRP and DAS28. Patients were then categorized as CD68NecrosisHIGH (CD68 + Necrosis ≥ 3) and CD68NecrosisLOW (CD68 + Necrosis < 3). CD68NecrosisHIGH exhibited higher pre-treatment disease activity [5.48 (1.6) versus 4.8 (1.7); p = 0.03] and a greater fall in DAS28 [1.99 (2.06) versus 1.1 (2.27), p = 0.03], SDAI [21.45 (IQR 23.3) versus 11.65 (IQR 17.5); p = 0.003] and CDAI [16 [14.9] versus 10.5 (20.1), p = 0.04]. CD68NecrosisHIGH patients had a higher EULAR Moderate/Good Response rate. CD68Necrosis score was incorporated into a probability matrix model together with clinical features (SJC44 and DAS28) to predict achieving a Moderate/Good EULAR Response Criteria at 3 months with a good performance (AUC 0.724).ConclusionFN and CD68 + in ERA synovial biopsies identify patients with higher disease activity and predict a better treatment response at three months. A model including synovial CD68 and fibrinoid necrosis with baseline clinical features predicts EULAR response at 3 months.

Optimal Network Pairwise Comparison

We are interested in the problem of two-sample network hypothesis testing: given two networks with the same set of nodes, we wish to test whether the underlying Bernoulli probability matrices of the two networks are the same or not. We propose Interlacing Balance Measure (IBM) as a new two-sample testing approach. We consider the Degree-Corrected Mixed-Membership (DCMM) model for undirected networks, where we allow severe degree heterogeneity, mixed-memberships, flexible sparsity levels, and weak signals. In such a broad setting, how to find a test that has a tractable limiting null and optimal testing performances is a challenging problem. We show that IBM is such a test: in a broad DCMM setting with only mild regularity conditions, IBM has N ( 0 , 1 ) as the limiting null and achieves the optimal phase transition. While the above is for undirected networks, IBM is a unified approach and is directly implementable for directed networks. For a broad directed-DCMM (extension of DCMM for directed networks) setting, we show that IBM has N ( 0 , 1 / 2 ) as the limiting null and continues to achieve the optimal phase transition. We have also applied IBM to the Enron email network and a gene co-expression network, with interesting results.

ANALYSIS OF EXPORTS OF FIVE MAIN FISHERY PRODUCT COMMODITIES IN BATAM CITY USING THE MARKOV CHAIN MODEL

Indonesia, an archipelagic country, boasts significant potential in the fisheries sector, notably in the export of fishery products. Batam as a free trade zone plays a major role in Indonesia's fishery product exports. According to the Central Statistics Agency's data, there has been a 2.47% decline in fish and shrimp exports in January and February 2023 compared to January 2022. Therefore, further analysis is needed on projected exports of fishery products in Batam. This study focuses on the utilization of the Markov chain model to analyze the export dynamics of Batam City's primary fishery product commodities, including Seaweed, Fish, Shrimp, Crab, and Shellfish. Monthly export data for Batam City's fishery products throughout 2021-2023 sourced from The Ministry of Marine Affairs and Fisheries. Based on these data, the transition probability matrix will be calculated based on the number of export transitions below and above the Moving Average (MA 12) value. The Markov chain model for this case is an erdogic Markov chain model, in order that the limiting probability of the Markov chain model can be determined. The result shows that exports of fish, shrimp and crab in the long term will exceed the average export size in the past year each after 7 months, 11 months, and 14 months later respectively, while exports of seaweed and shellfish in the long term will be below the large average exports in the past year after 15 and 5 months. This research result offer valuable insights for the government to formulate strategies and policies concerning fishery product exports

Spectral ensemble clustering with doubly stochastic co-association matrix

Although co-association (CA) matrix-based spectral ensemble clustering (SEC) has achieved significant success, it faces two persistent challenges: (1) CA matrix-based SEC may result in imbalanced cluster outcomes, and (2) SEC encounters difficulties due to the high time and space complexity resulting from the quadratic scaling of the CA matrix with the number of data points. To tackle these issues, we propose a doubly stochastic co-association (DSCA) matrix construction method. We have theoretically demonstrated that DSCA matrix-based SEC can achieve a more balanced cluster outcome compared to CA matrix-based SEC. Leveraging the DSCA matrix, we introduce two novel methods: Fast SEC (FSEC) and Approximate SEC (ASEC). FSEC efficiently solves the normalized cut on the DSCA matrix as it is symmetric and doubly stochastic. In ASEC, we employ random samples rather than the entire big dataset for ensemble clustering. After obtaining the ensemble clustering result of benchmark random sample, the approximate ensemble clustering result for all data points is derived using our newly proposed probability nearest neighbors (PNN) algorithm. Experimental results on both real-world and synthetic datasets are evaluated in terms of accuracy (ACC), normalized mutual information (NMI), purity, F1-score, and adjusted rand index (ARI) and confirm the scalability and effectiveness of the proposed FSEC and ASEC.

Demystifying the Karpelevič theorem

The statement of the Karpelevič theorem concerning the location of the eigenvalues of stochastic matrices in the complex plane (known as the Karpelevič region) is long and complicated and his proof methods are, at best, nebulous. Fortunately, an elegant simplification of the statement was provided by Ito—in particular, Ito's theorem asserts that the boundary of the Karpelevič region consists of arcs whose points satisfy a polynomial equation that depends on the endpoints of the arc. Unfortunately, Ito did not prove his version and only showed that it is equivalent.More recently, Johnson and Paparella showed that points satisfying Ito's equation belong to the Karpelevič region. Although not the intent of their work, this initiated the process of proving Ito's theorem and hence providing another proof of the Karpelevič theorem.The purpose of this work is to continue this effort by showing that an arc appears in the prescribed sector. To this end, it is shown that there is a continuous function λ:[0,1]⟶C such that PI(λ(α))=0, ∀α∈[0,1], where PI is a Type I reduced Ito polynomial. It is also shown that these arcs are simple. Finally, an elementary argument is given to show that points on the boundary of the Karpelevič region are extremal whenever n>3.

Comparison of Fuzzy Grey Markov Model (1,1) and Fuzzy Grey Markov Model (2,1) in Forecasting Gold Prices in Indonesia

Currently, gold investment is considered promising despite the ever-changing price of gold. However, obtaining optimal profits is a challenge for investors. Therefore, a proper forecasting method is needed to forecast the gold price so investors can know the best transaction time. This study used two forecasting methods: the Fuzzy Grey Markov Model (1,1) and a new, never-before-used approach, the Fuzzy Grey Markov Model (2,1). The Fuzzy Grey Markov Model (2,1) approach is interesting because it can be considered for forecast data that shows varying increases and decreases, such as the gold price data used in this study. Both methods are combined models that utilize fuzzy logic to handle uncertainty in data; the Grey model forms a forecasting model, and the Markov chain determines the state transition probability matrix. Next, the error rates of the two methods are compared based on the Mean Absolute Percentage Error (MAPE) value to obtain the best forecasting method. As a result of this study, the Fuzzy Grey Markov Model (1,1) was chosen as the best forecasting method with a MAPE value of 0.28%.

Random walk algorithms for solving nonlinear chemotaxis problems

Abstract Random walk based stochastic simulation methods for solving a nonlinear system of coupled transient diffusion and drift-diffusion equations governing a two-component chemotaxis process are developed. The nonlinear system is solved by linearization, the system is evolved in time, by small time steps, where on each step a linear system of equations is solved by using the solution from the previous time step. Three different stochastic algorithms are suggested, (1) the global random walk on grid (GRWG), (2) a randomized vector algorithm (RVA) based on a special transformation of the original matrix to a stochastic matrix, and (3) a stochastic projection algorithm (SPA). To get high precision results, these methods are combined with an iterative refinement method.